Bulletin of the Korean Mathematical Society (대한수학회보)

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CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS

  • Li, Zhongping (College of Mathematic and Information China West Normal University) ;
  • Mu, Chunlai (College of mathematics and Statistics Chongqing University) ;
  • Du, Wanjuan (College of Mathematic and Information China West Normal University)
  • Received : 2011.05.12
  • Published : 2013.01.31
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Abstract

In this paper, we consider the positive solution to a Cauchy problem in $\mathbb{B}^N$ of the fast diffusive equation: ${\mid}x{\mid}^mu_t={div}(\mid{\nabla}u{\mid}^{p-2}{\nabla}u)+{\mid}x{\mid}^nu^q$, with nontrivial, nonnegative initial data. Here $\frac{2N+m}{N+m+1}$ < $p$ < 2, $q$ > 1 and 0 < $m{\leq}n$ < $qm+N(q-1)$. We prove that $q_c=p-1{\frac{p+n}{N+m}}$ is the critical Fujita exponent. That is, if 1 < $q{\leq}q_c$, then every positive solution blows up in finite time, but for $q$ > $q_c$, there exist both global and non-global solutions to the problem.

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References

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